Question Paper: Engineering Maths 3 : Question Paper Dec 2015 - Computer Engineering (Semester 4) | Pune University (PU)
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## Engineering Maths 3 - Dec 2015

### Computer Engg (Semester 4)

TOTAL MARKS: 100
TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.
(2) Attempt any four from the remaining questions.
(3) Assume data wherever required.
(4) Figures to the right indicate full marks.

### Solve any one question from Q1 and Q2

1 (a) (i) (D2-2D-3) y=3e-3x sin e-3x + cos (e-3x).(4 marks) 1 (a) (ii) (D2-2D+2) y=ex tan x. (By variation of parameters).(4 marks) 1 (a) (iii) $$x^2 \dfrac {d^2 y}{dx^2}-2x \dfrac {dy}{dx} - 4y=x^2$$(4 marks) 1 (b) Find the Fourier transform of e-|x| and hence show that: $$\int^{\infty}_{-\infty} \dfrac {e^{i\lambda x}}{1+\lambda^2} d \lambda = \pi e^{-|x|}$$(4 marks) 2 (a) An unchanged condenser of capacity C charged by applying an e.m.f. of value $\dfrac {t}{\sqrt{LC}}$ through the LEDs of inductance L and of negligible resistance. The charge Q on the place of condenser satisfied the differential equation: $$\dfrac {d^2Q}{dt^2} + \dfrac {Q}{LC} = \dfrac {E}{L}\sin \dfrac {t}{\sqrt{LC}}$$ Prove that the charge at any time t is given by: $$Q= \dfrac {EC}{2} \left [ \sin \dfrac {t}{\sqrt{LC}} - \dfrac {t}{\sqrt{LC}} \cos \dfrac {t}{\sqrt{LC}} \right ]$$(4 marks) 2 (b) Find the Inverse Z-transform (any one): $$i) \ F(z) = \dfrac {z+2}{z^2 - 2z+1} \ \text {for }|z|>1. \\ ii) \ F(z) = \dfrac {10z}{(z-1)(z-2)} \ \text {(Use inversion integral method).}$$(4 marks) 2 (c) Solve the following difference equation to find {f(k)}: $$f(k+1)+ \dfrac {1}{4} f(k) = \left ( \dfrac {1}{4} \right )^k, \ k\ge 0, \ f(0)=0$$(4 marks)

### Solve any one question from Q3 and Q4

3 (a) The first four moments of distribution about the value 4 are -1.5, 17, -30 and 108. Obtain the first four central moments, mean, standard deviation and coefficient of skewness and kurtosis.(4 marks) 3 (b) If the probability that a concrete cube fails is 0.001. Determine the probability that out of 1000 cubes:
i) exactly two
ii) more than one cubes will fail.
(4 marks)
3 (c) Show that: $$\overline F= ( y \sin z - \sin x) \overline i (x \sin z +2 yz)\overline j + (xy\cos z+y^2)\overline k$$ is irrotational and hence find scalar function ϕ s.t. F=∇ϕ.(4 marks) 4 (a) Find the directional derivative of ϕ=4xz3-3x2y2z at (2, -1, 2) along a line equally inclined with co-ordinate axes.(4 marks) 4 (b) For a solenoidal vector field F, show that:
curl curl curl curl F=∇4F.
(4 marks)
4 (c) The regression equations are:
8x+10y+66=0 and 40x-18y=214.
The value of variance of x is 9. Find.
i) The mean values of x and y
ii) The correlation coefficient between x and y
iii) The standard deviation of y.
(4 marks)

### Solve any one question from Q5 and Q6

5 (a) Find the work done in moving a particle once round the ellipse: $$\dfrac {x^2}{16}+ \dfrac {y^2}{4}=1, \ z=0$$ under the field of force given by: $$\overline{F} = (2x-y+z)\overline i + (x+y-z^2) \overline j + (3x-2y + 4z) \overline{k}$$(4 marks) 5 (b) Evaluate: $$\iint_s (\nabla \times \overline{F} ) \cdot \widehat{n} \ dS$$ where $$\overline F = (x^3 - y^3) \overline {i} - xyz \overline j + y^2 \overline {k}$$ and S is the surface x2+4y2+z2-2x=4 above the plane x=0.(4 marks) 5 (c) Evaluate: $$\iint_s \overline {F} \cdot \overline {dS}$$ using divergence theorem, where $$\overline F= x^3 \overline i + y^3 \overline j+z^3 \overline k$$ and S is the surface of sphere x2+y2+z2=a2.(5 marks) 6 (a) If $\overline F = x^2 \overline i + (x-y) \overline j + (y+z) \overline k$ displaces a particle from A(1, 0, 1) to B(2, 1, 2) along the straight line AB, find work done.(4 marks) 6 (b) Evaluate: $$\int_C (e^x dx + 2ydy - dx)$$ where C is the curve x2+y2=4, z=2.(4 marks) 6 (c) Evaluate: $$\int_s \overline F \cdot \overline {dS}$$ using Gauss divergence theorem, where: $$\overline F = 2xy\overline i + yz^2 \overline j + xz\overline k$$ and S is the region bounded by:
x=0, y=0, z=0, y=3, x+2z=6.
(5 marks)

### Solve any one question from Q7 and Q8

7 (a) Show that u=y3-3x2y is harmonic function. Find its harmonic conjugate and the corresponding analytic function f(z) in terms of z.(5 marks) 7 (b) Using Cauchy's integral formula, evaluate: $$\int_C \dfrac {2z^2 + z +5}{(z-3 /2)^2}dz$$ where C is $\dfrac {x^2}{4} + \dfrac {y^2}{9} = 1.$(4 marks) 7 (c) Find the bilinear transformation which maps the points z=1, i, -1, onto the points w=0, 1, ∞.(4 marks) 8 (a) If f(z) is an analytic function v2=u, then show that f(z) is constant function.(4 marks) 8 (b) Using residue theorem evaluate: $$\int_C \dfrac {z}{z^4 +13z^2 + 36} dz$$ where 'C' is the circle $z| = \dfrac {5}{2}.$(5 marks) 8 (c) Find the map of the circle |z=i|=1 under the transformation $w=\dfrac {1}{w}$ into w-plane.(4 marks)